To factorize the polynomial [tex]\( x^2 + 27x + 162 \)[/tex], we need to express it in the form of [tex]\((x + a)(x + b)\)[/tex].
We start by identifying the terms [tex]\(a\)[/tex] and [tex]\(b\)[/tex] which satisfy the following conditions:
1. The product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] should be equal to the constant term, which is 162.
2. The sum of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] should equal the coefficient of the linear term, which is 27.
Let's identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy these conditions:
1. The possible pairs (a, b) such that [tex]\(ab = 162\)[/tex] are:
- (1, 162)
- (2, 81)
- (3, 54)
- (6, 27)
- (9, 18)
- (9, 18)
2. From these pairs, we need to find one where the sum [tex]\(a + b\)[/tex] equals 27:
- [tex]\(1 + 162 = 163\)[/tex]
- [tex]\(2 + 81 = 83\)[/tex]
- [tex]\(3 + 54 = 57\)[/tex]
- [tex]\(6 + 27 = 33\)[/tex]
- [tex]\(9 + 18 = 27\)[/tex]
Among these pairs, the one which gives us [tex]\(a + b = 27\)[/tex] is (9, 18).
Thus, the polynomial can be factored as:
[tex]\[ (x + 9)(x + 18) \][/tex]
The terms completing the factorization model are therefore:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
x & x^2 & \\
\hline
18 & (x + 9) & 162 \\
\hline
\end{tabular}
\][/tex]
Clearly, the correct answer from the given options is:
[tex]\[
27,\, 9x,\, 18x
\][/tex]
So, the correct factors are [tex]\((x + 9)\)[/tex] and [tex]\((x + 18)\)[/tex].
